logarithmic derivative


Given a function f, the quantity f/f is known as the logarithmic derivativeMathworldPlanetmath of f. This name comes from the observation that, on account of the chain ruleMathworldPlanetmath,

(logf)=flog(f)=f/f.

The logarithmic derivative has several basic properties which make it useful in various contexts.

The logarithmic derivative of the product of functions is the sum of their logarithmic derivatives. This follows from the product ruleMathworldPlanetmath:

(fg)fg=fg+fgfg=ff+gg

The logarithmic derivative of the quotient of functions is the difference of their logarithmic derivatives. This follows from the quotient ruleMathworldPlanetmath:

(f/g)f/g=fg-fgg2gf=ff-gg

The logarithmic derivative of the p-th power of a function is p times the logarithmic derivative of the function. This follows from the power ruleMathworldPlanetmathPlanetmath:

(fp)fp=pfp-1ffp=pff

The logarithmic derivative of the exponentialPlanetmathPlanetmath of a function equals the derivativePlanetmathPlanetmath of a function. This follows from the chain rule:

(ef)ef=effef=f

Using these identities, it is rather easy to compute the logarithmic derivatives of expressions which are presented in factored form. For instance, suppose we want to compute the logarithmic derivative of

ex2(x-2)3(x-3)x-1.

Using our identities, we find that its logarithic derivative is

2x+3x-2+1x-3-1x-1.
Title logarithmic derivative
Canonical name LogarithmicDerivative
Date of creation 2013-03-22 16:47:02
Last modified on 2013-03-22 16:47:02
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 11
Author rspuzio (6075)
Entry type Definition
Classification msc 26B05
Classification msc 46G05
Classification msc 26A24
Related topic ZeroesOfDerivativeOfComplexPolynomial